Tag Archives: Chris Doran

The Unintentional Obsfuscation of Physics

Sometimes it only takes one person's untimely demise to change history. There's an entire genre of literature that explores these possibilities, typically involving the biggest baddies of human history. The following video is an artful example that makes this point rather succinctly - while also leaving me profoundly uncomfortable (after all, it does involve the death of a child).

I am not aware of many examples of exploring alternative histories with regards to science, and by that I mean in more detail than what steampunk has to offer, although William Gibson and Bruce Sterling do a pretty good job of imagining a world in which Charles Babbage succeeded in introducing a mechanical computer to the world in their book "The Difference Engine".  The subject matter is certainly a worthwhile topic for another post , especially when contrasted with the challenges now to go beyond the Turing machine by getting Quantum Computing to the market. (h/t vznvzn)

william
William Kingdon Clifford (1845 – 1879). Had he lived longer physics would be taught differently.

The untimely death I am contemplating here is that of William Kingdon Clifford. If you are not immersed in physics and math, you have probably never heard his name, because we live in a world where he died young.

That meant it fell to Gibbs and Heaviside to clean up the Maxwell equations, which gave us the insufferable cross-product that confused leagues of students by requiring them to distinguish between polar and axial vectors.  It also meant that complex function theory got stuck in two dimensions, and that group theory was developed without the obvious geometric connection. Which in turn, once this approach started to take over, provoked older physicists, such as Schrödinger, to coin the term "Gruppenpest" (group pestilence). It also created a false symmetry between the electric and magnetic fields, motivating the quest for the ever elusive magnetic monopol. Last but not least, it led to the confused notion that spin is an intrinsically quantum mechanical property, something that is still taught in universities across the globe to this day.

It's impossible to do Geometric Algebra (GA) justice in one short blog post, but David Hestenes managed to do so in a fairly concise and highly readable paper, the 2002 Oersted Medal Lecture.

It is hard to overstate the profound effect this paper had on me.  The only thing it compares to is when I first learned of Euler's formula many years ago in my first physics semester.  And the similarities are striking, not only due to the power of bringing together seemingly disparate areas of mathematics by putting them into a geometric context. In the latter case, the key is the imaginary unit, which was originally introduced to solve for negative square roots, and thus allows for the fundamental theorem of algebra. In fact, it turns out that complex numbers can be neatly embedded into geometric algebra and are isomorphic to the 2d GA case. Also, Quaternion are part of the 3d geometric algebra and have a similarly satisfying geometric interpretation.

All this is accomplished by introducing a higher level concept of vector.  For instance, rather than using a cross product, an outer product is defined that creates a bivector that can be thought of as a directed plane segment.

Hestenes makes a convincing case that geometric algebra should be incorporated into every physics curriculum. He wrote some excellent textbooks on the subject, and thankfully, numerous other authors have picked up the mantle (outstanding is John W. Arthur's take on electrodynamics and Chris Doran's ambitious and extensive treatment).

The advantages of geometric algebra are so glaring and the concepts so natural that one has to wonder why it took a century to be rediscovered.  John Snygg puts it best in the preface to his textbook on differential geometry:

Although Clifford was recognized worldwide as one of England’s most distinguished mathematicians, he chose to have the first paper published in what must have been a very obscure journal at the time. Quite possibly it was a gesture of support for the efforts of James Joseph Sylvester to establish the first American graduate program in mathematics at Johns Hopkins University. As part of his endeavors, Sylvester founded the American Journal of Mathematics and Clifford’s first paper on what is now known as Clifford algebra appeared in the very first volume of that journal.

The second paper was published after his death in unfinished form as part of his collected papers. Both of these papers were ignored and soon forgotten. As late as 1923, math historian David Eugene Smith discussed Clifford’s achievements without mentioning “geometric algebra” (Smith, David Eugene 1923). In 1928, P.A.M. Dirac reinvented Clifford algebra to formulate his equation for the electron. This equation enabled him to predict the discovery of the positron in 1931. (...)

Had Clifford lived longer, “geometric algebra” would probably have become mainstream mathematics near the beginning of the twentieth century. In the decades following Clifford’s death, a battle broke out between those who wanted to use quaternions to do physics and geometry and those who wanted to use vectors. Quaternions were superior for dealing with rotations, but they are useless in dimensions higher than three or four without grafting on some extra structure.

Eventually vectors won out. Since the structure of both quaternions and vectors are contained in the formalism of Clifford algebra, the debate would have taken a different direction had Clifford lived longer. While alive, Clifford was an articulate spokesman and his writing for popular consumption still gets published from time to time. Had Clifford
participated in the quaternion–vector debate, “geometric algebra” would have received more serious consideration.